Maximal Brill-Noether loci via degenerations and double covers

Abstract

Using limit linear series on chains of curves, we show that closures of certain Brill--Noether loci contain a product of pointed Brill--Noether loci of small codimension. As a result, we obtain new non-containments of Brill--Noether loci, in particular that all dimensionally expected non-containments hold for expected maximal Brill--Noether loci. Using these degenerations, we also give a new proof that Brill--Noether loci with expected codimension $-\ddot{I}\le \lceil g/2\rceil$ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill--Noether loci by considering the locus of the source curves of unramified double covers.

16 pages, accepted version, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)